Data Sufficiency - 51

A store purchases 20 coats that each cost an equal amount and then sold each of the 20 coats at an equal price, what was the store's gross profit on the 20 coats?

1). If the selling price per coat had been twice as much, the store's gross profit on the 20 coats would have been 2400

2). If the store selling price per coat had been $2 more, the store's gross profit on the 20 coats would have been 440

Answer: B

Suppose the cost of each coat = x

Suppose sales price = y

=> Total cost = 20x and sales price = 20y

We are required to find the profit: 20(y-x).

From statement 1): It is given that 20(2y-x)=2400

Doesn't helps to find out 20(y-x) --- insufficient

From statement 2): It is given that 20(y+2-x) = 440,

=> 20(y-x) = 400 --- sufficient

Hence B

Problem Solving - 56

A certain stock exchange designates each stock with a one- , two-,or three-letter code ,where each letter is selected from the 26 letters of the alphabet. If the letter may be repeated and if the same letters used in a different order constitute a different code, how many different stocks is it possible to uniquely designate with these codes?

A) 2951

B) 8125

C) 15600

D) 15302

E) 18278

Answer: E

Total ways to pick a 3 letter code = 26*26*26 = 26^3
Total ways to pick a 2 letter code = 26*26 = 26^2
Total ways to pick a 1 letter code = 26

Thus total stocks: 26^3 + 26^2 + 26 = 26(26^2 + 26 + 1) = 18278

Problem Solving - 54

This year Henry will save a certain amount of his income, and he will spend the rest. Next year Henry will have no income, but for each dollar that he saves this year, he will have 1 + r dollars available to spend. In terms of r, what fraction of his income should Henry save this year so that next year the amount he was available to spend will be equal to half the amount that he spends this year?

(A) 1/(r+2)
(B) 1/2(r+2)
(C) 1/(3r+2)
(D) 1/(r+3)
(E) 1/(2r+3)


Answer: E

Let total income = x
Let saved = y
=> spent = x-y

Total dollars available that he can spend next year = y(1+r)

Given y(1+r) = (x-y)/2
=> y = x/(2r+3)

Fraction (y/x) = 1/(2r+3)

Hence E

Problem Solving - 53

A student worked 20 days. For each of the amount shown (see attached table) in the first row of the table, second row gives the number of days the student earned that amount. Median amount of money earned per day for 20 days is?

A) 96
B) 84
C) 80
D) 70
E) 48

Answer: B

Median day = 20+1)/2 = 10.5 th -- money earned was 84
= Average value of 10th and 11th day in the sequence = Median amount of money
Average value of 10th day = 84
Average value of 11th day = 84
Average value of 10th and 11th day = 84 ans

Data Sufficiency - 50

What is the median number of employees assigned per project for the projects at Company Z?

(1) 25 percent of the projects at Company Z have 4 or more employees assigned to each project.
(2) 35 percent of the projects at Company Z have 2 or fewer employees assigned to each project.

Answer: OA - C

From Statement 1): It is given that 25 percent of the projects at Company Z have 4 or more employees assigned to each project - but we donot know the percentage of projects who have employees less than 4 or in other words we do not have any information about the rest 75% projects ---- hence insufficient

From Statement 2): It is given that 35 percent of the projects at Company Z have 2 or fewer employees assigned to each project - but we donot know the percentage of projects who have employees more than 2 or in other words we do not have any information about the rest 65% projects---- hence insufficient

Taking both the statements together:
25 percent of the projects at Company Z have employees 4 , 5, 6..
35 percent of the projects at Company Z have employees 2, 1, 0
=> 40% of projects have 3 employees = > median value is 3

(1-35)employees -- (36-75)employees -- (76-100)employees
2 or less than 2 ---------3, 3, 3, ------------ 4 or more than 4

Problem Solving - 52

A boat traveled up stream a distance of 90 miles at an average speed of (v-3) mph and then traveled the same distance downstream at an average speed of (v+3) mph. If the trip upstream took half an hour longer than the trip downstream, how many hours did it take the boat to travel downstream?

(A) 2.5
(B) 2.4
(C) 2.3
(D) 2.2
(E) 2.1

Answer: A

Total upstream time taken by boat to travel = 90/ (v-3) hrs
Total downstream time taken by boat to travel = 90/ (v+3) hrs

It is given that upstream took half an hour longer than the trip downstream:

=> 90/(v-3) - 90/(v+3) = 1/2
=> [90*(v+3) - 90(v-3)] / [(v^2) -9)] = 1/2
=> 90*[(v+3)-(v-3)] / [(v^2) -9] = 1/2
=> 90*6/ [(v^2) -9] = 1/2
=> v^2 - 9 = 2*90*6
=> v^2 = 2*90*6 + 9 = 2*9*10*6 + 9 = 9(2*10*6+1) = 9(121) = 9*11*11
=> v = 3*11 = 33

We are suppose to calculate = 90/(v+3) = 90/36 = 30/12 = 5/2 = 2.5 ans

Problem Solving - 51

Each of the 10 machines works at the same constant rate of doing certain job. The amount of time needed by 10 machines, working together to complete the job is 16 hrs. How many hours are needed if only 8 machines working together were to complete the job?

A. 18
B. 20
C. 22
D. 24
E. 26

Answer: B

10 machines, working at same constant rate, take time to complete a job = 16 hrs
thus 1 machine takes time to complete a job = 10 * 16 = 160 hrs
=> 8 machines will take = 160/8 = 20 hrs to complete the job.

Problem Solving - 49

Pumps A, B, and C operate at their respective constant rates. Pumps A and B, operating simultaneously, can fill a certain tank in 6/5 hours, pumps A and C, operating simultaneously, can fill the tank in 3/2 hours; and pumps B and C, operating simultaneously, can fill the tank in 2 hours. How many hours does it take pumps A, B, and C, operating simultaneously, to fill the tank?

A) 1/3
B) 1/2
C) 2/3
D) 5/6
E) 1

Answer: E

1/A + 1/B = 5/6
1/B +1/C = 2/3
1/C + 1/A =1/2

Adding  all three equations above we get:

1/A + 1/B + 1/B +1/C + 1/C + 1/A = 5/6 + 2/3 + 1/2
2(1/A + 1/B + 1/C) = (5 + 4 + 3)/ 6 = 12/6
1/A + 1/B + 1/C = 12/6*2 = 1

=> Pumps A, B, and C, operating simultaneously will fill the tank in 1 hr

Problem Solving - 47

If n is a positive integer and the product of all the integers from 1 to n, inclusive, is a multiple of 990, what is the least possible value of n?

A) 10
B) 11
C) 12
D) 13
E) 14

Answer: B

990 is a multiple of n! implies it must contain all the prime factors of 990
Largest prime factor of 990 is 11 implies n! must have 11 as a factor

Now since n! = 990x where x is integer it implies it can have prime factors more than 11 but not less than 11

Thus least possible value of n is thus 11

Problem Solving - 46

A certain restaurant offers 6 kinds of cheese and 2 kinds of fruit for its dessert platter. If each dessert platter contains an equal number of kinds of cheese and kinds of fruit, how many different dessert platters could the restaurant offer?

A) 8
B) 12
C) 15
D) 21
E) 27

Answer: E

Kinds of platter:

1 cheese + 1 fruit
Total = 6 * 2 = 12 types of platters

2 cheese + 2 fruit
Total = 6C2 * 2C2 = 15 * 1 = 15 types

Total: 12 + 15 = 27

Data Sufficiency - 43

In the figure shown, what is the value of x?

(1) The length of line segment of QR is equal to the length of line segment RS
(2) The length of line segment of ST is equal to the length of line segment TU

Answer: C

From statement (1): Length of line segment of QR is equal to the length of line segment RS ..this implies angle RQS = angle RSQ = p(say)
From statement (2): Length of line segment of ST is equal to the length of line segment TU .. this implies angle TUS = angle TSU = q(say)

Hence p+p+angle QRS = 180 --- eq(1) and q+q+angle UTS = 180 --- eq(2)
Thus, p+q+x = 180

Now because angle RPT = 90, QRS+UTS= 90
adding eq(1) and eq(2) we get:
2p+2q+QRS+UTS = 360
2p+2q+90=360
p+q = 270/2 = 135

Now x = 180-p-q..hence the answer C
x = 180 - (p+q) = 180 - 135 = 45

Data Sufficiency - 40

In XY plane, does the line with equation y=3x+2 contain point (r,s)?

1) (3r + 2 - s)(4r + 9 - s) = 0
2) (4r - 6 - s)(3r + 2 - s) = 0

Answer: C

Given that y = 3x+2 implies that does 3x+2-y = 0 contains the point (r,s) implies is (3r+2-s) = 0 ?

From statement (1): (3r+2-s)(4r+9-s) = 0 implies either (3r+2-s) = 0 or (4r+9-s) = 0.
Now when (3r+2-s)...the line passes through (r,s)
When (4r+9-s) = 0 ...we cannot determine that whether the line passes through (r,s) or not.
Hence insufficient

From statement (2): (4r-6-s)(3r+2-s) = 0 implies either (4r-6-s) = 0 or (3r+2-s) = 0
Now when (4r-6-s) = 0 ... we cannot determine that whether the line passes through (r,s) or not
When (3r+2-s) = 0..the line passes through (r,s)
Hence insufficient

Taking statement (1) and (2) together: (3r+2-s)(4r+9-s) = 0 and (4r-6-s)(3r+2-s) =0... We cannot have both 4r+9-s=0 and 4r-6-s=0 so it is (3r+2-s) = 0 ... only this equation makes both the equations to be 0
Hence sufficient

Problem Solving - 41

If eleven consecutive integers are listed from least to greatest, what is the average (arithmetic mean) of the eleven integers?

(1) The average of the first nine integers is 7.
(2) The average of the last nine integers is 9.

Answer: D

Let the numbers be a, b, c, d, e, f, g, h , i, j, k

i) For odd number of consecutive integers median = mean
ii)We also know that the median is the "middle" number in a group (when arranged in ascending or descending order) consisting of an odd number of numbers

We have to find f

From statement (1): It is given that average of first nine numbers = 7
Hence this implies e = 7 ..since it is given numbers are consecutive hence f = 8
Thus sufficient

From statement (2): It is given that average of last nine numbers = 9
Hence this implies g = 9..since it is given numbers are consecutive hence f = 8
Thus sufficient

Data Sufficiency - 39

If *triangle* denotes one of the four arithmetic operations addition, subtraction, multiplication and division, what is the value of 1 *triangle* 2 ?

(1) n *triangle* 0 = n for all integers n.
(2) n *triangle* n = 0 for all integers n.

Answer: B

From statement (1): *triangle* can be both positive or negative as

n-0 = n
n+0 = n

Hence insufficient

From statement (2): *triangle* can only be negative in this case as

n-n = 0

Hence sufficient

Problem Solving - 40

5 pieces of wood have an average (arithmetic mean) length of 124 centimeters and a median length of 140 centimeters. What is the maximum possible length in centimeters of the shortest piece of wood?

A. 90

B. 100

C. 110

D. 130

E. 140

Answer: B

Shortest to Longest length ---- L1, L2, L3 = 140, L4, L5
L1 + L2 + 140 + L4 + L5 = 5 * 124
For L1 to be the maximum, L4 and L5 should be minimum

L1=L2
Thus, L1 + L2 = (5*124) - (140*3) = 620 - 420 = 200

Hence L1 = L2 = 100

Problem Solving - 39

On a certain day, Tim invest $1,000 at 10 percent annual interest, compound annually, and Lana invested $ 2,000 at 5 percent annual interest, compound annually. The total amount of interest earned by Tim's investment in the first 2 years was how much greater than the total amount of interest earned by Lana's investment in the first 2 years?

A. 5
B. 15
C. 50
D. 100
E. 105

Answer: A

Amount = P[1+(r/100)] ^ t
Tim's investment Amount = 1000(1+10%)^2=1210
Interest earned by Tim = Amount - P = 1210 - 1000 = 210
Lana's investment Amount = 2000(1+5%)^2=2205
Interest earned by Lana = Amount - P = 2205 - 2000 = 205
Hence 210-205=5

Problem Solving - 38

6 machines, each working at the same constant rate, together can complete a certain job in 12 days, How many additional machines, each working at the same constant rate, will be needed to complete the job in 8 days?

A. 2

B. 3

C. 4

D. 6

E. 8

Answer: B

6 machines take = 12 days.
Therefore 1 machine = 12*6 days.
8 days will take = (12*6)/8 = 9 machines.

No of additional machines required = 9-6 = 3

Problem Solving - 37

Of the 800 companies in Company X, 70% have been with the company for at least 10 years. If y of these "long-term" members were to retire, and no other employee changes were to occur, what value of y would reduce the percent of "long-term" employees in the company to 60%.

A) 200
B) 160
C) 112
D) 80
E) 56

Answer: A

Assume y=x
The number of people working more than 10 years = 70% of 800 = 560
Hence (560-x)/(800-x)=60%
Thus x=200

OR

The number of long term workers = 70% of 800 = 560
Now if y of the long term workers retired, then long term workers left are 560-y and the number of total employees = 800-y

Thus (560-y)/(800-y) * 100 = 60
560-y = 480 - 0.6y
80 = 0.4y
y = 200 Ans

Problem Solving - 36

Equal amounts of water were poured into two empty jars of different capacities, which made one jar 1/4 full and the other jar 1/3 full. If the water in the jar with lesser capacity is then poured into the jar with the greater capacity, what fraction of the larger jar will be filled with water?

A. 1/7

B. 2/7

C. 1/2

D. 7/12

E. 2/3

Answer: C

The jar that is 1/3 full is smaller
water in jar 1 = water in jar 2
jar 1 is now twice as full (1/4)*2 = 1/2

or

1/4 + 1/4 = 1/2

Problem Solving - 35

The rate of a certain chemical reaction is directly proportional to the square of the concentration of chemical A present and inversely proportional to the concentration of chemical B present. If the concentration of chemical B is increased by 100 percent, which of the following is closest to the percent change in the concentration of chemical A required to keep the reaction rate unchanged?

A. 100% decrease
B. 50% decrease
C. 40% decrease
D. 40% increase
E. 50% increase

Answer: D

rate = k*(A1^2)/B
If concentration of chemical B is increased by 100 percent then
rate = k*(A2^2)/2B

(A2/A1) ^ 2 = 2
A2/A1 = Square root (2)
(A2/A1) - 1 = Square root (2)-1 = 0.414
(A2-A1)/A1 = 0.414 = 40% approximately